Lower Bound on Weights of Large Degree Threshold Functions

Abstract

Abstract. An integer polynomial p of n variables is called a threshold gate for a Boolean function f of n variables if for all x ∈ {0, 1} n f (x) = 1 if and only if p(x) 0. The weight of a threshold gate is the sum of its absolute values.In this paper we study how large a weight might be needed if we fix some function and some threshold degree. We prove 2 Ω(2 2n/5 ) lower bound on this value. The best previous bound was 2In addition we present substantially simpler proof of the weaker 2This proof is conceptually similar to other proofs of the bounds on weights of nonlinear threshold gates, but avoids a lot of technical details arising in other proofs. We hope that this proof will help to show the ideas behind the construction used to prove these lower bounds.

The Characterizations of Hyper‐Star Graphs Induced by Linearly Separable Boolean Functions

“…Many problems related to Boolean functions have been widely investigated. For instance, Podolskii [25,26] studied the weights of large degree linearly separable Boolean functions. Partitions of a discrete set of points in a real space by parallel hyperplanes are also researched intensively [27−29] .…”